Page 296 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 11] INFINITE SERIES 287

1 n 1
X 1 x þ 2 X 1
11.26. For what values of x does (a) ; ðbÞ converge?
2n 1 x 1
n¼1 n¼1 ðx þ nÞðx þ n 1Þ
n
1 x þ 2 2n 1 x þ 2 x þ 2

: Then lim u nþ1 ¼ lim if x 6¼ 1; 2:
2n 1 x 1 n!1 u n n!1 2n þ 1 x 1 x 1
ðaÞ u n ¼ ¼

x þ 2 x þ 2 x þ 2
Then the series converges if < 1, diverges if > 1, and the test fails if ¼ 1, i.e.,
x 1 x 1 x 1
1
x ¼ .
2
If x ¼ 1the series diverges.
If x ¼ 2the series converges.
n
1
X
1
If x the series is 2n 1 which converges.
ð 1Þ
2
n¼1
1
1
x þ 2
Thus, the series converges for < 1, x ¼ and x ¼ 2, i.e., for x @ .
x 1
2 2

1

(b) The ratio test fails since lim u nþ1 ¼ 1, where u n ¼ : However, noting that

n!1 u n
ðx þ nÞðx þ n 1Þ
1 1 1
x þ n 1 x þ n
¼
ðx þ nÞðx þ n 1Þ
we see that if x 6¼ 0; 1; 2; ... ; n,

1 1 1 1 1 1
S n ¼ u 1 þ u 2 þ þ u n ¼ þ þ þ
x x þ 1 x þ 1 x þ 2 x þ n 1 x þ n
1 1
x x þ n
¼
and lim S n ¼ 1=x,provided x 6¼ 0; 1; 2; 3; .. . .
n!1
Then the series converges for all x except x ¼ 0; 1; 2; 3; ... ; and its sum is 1=x.
UNIFORM CONVERGENCE
2
11.27. Find the domain of convergence of ð1 xÞþ xð1 xÞþ x ð1 xÞþ .
Method 1:
2
Sum of first n terms ¼ S n ðxÞ¼ ð1 xÞþ xð1 xÞþ x ð1 xÞþ þ x n 1 ð1 xÞ
2
2
¼ 1 x þ x x þ x þ þ x n 1 x n
¼ 1 x n
n
If jxj < 1, lim S n ðxÞ¼ lim ð1 x Þ¼ 1.
n!1 n!1
If jxj > 1, lim S n ðxÞ does not exist.
n!1
If x ¼ 1; S n ðxÞ¼ 0 and lim S n ðxÞ¼ 0.
n!1
n
If x ¼ 1; S n ðxÞ¼ 1 ð 1Þ and lim S n ðxÞ does not exist.
n!1
Thus, the series converges for jxj < 1 and x ¼ 1, i.e., for 1 < x @ 1.
Method 2,using the ratio test.
The series converges if x ¼ 1. If x 6¼ 1 and u n ¼ x n 1 ð1 xÞ,then lim u nþ1 ¼ lim jxj.

n!1 u n n!1
Thus, the series converges if jxj < 1, diverges if jxj > 1. The test fails if jxj¼ 1. If x ¼ 1, the series
converges; if x ¼ 1, the series diverges. Then the series converges for 1 < x @ 1:
11.28. Investigate the uniform convergence of the series of Problem 11.27 in the interval
1
1
1
1
(a) < x < ,(b) @ x @ , ðcÞ :99 @ x @ :99; ðdÞ 1 < x < 1,
2 2 2 2
ðeÞ 0 @ x < 2.